Sunday, 15 September 2013

Sum of a Geometric series and its Application

Sum

The sum
of a geometric series is finite as long as the terms approach zero; as
the numbers near zero, they become insignificantly small, allowing a sum
to be calculated despite the series being infinite. The sum can be
computed using the self-similarity of the series.

Example

Consider the sum of the following geometric series:

This series has common ratio 2/3. If we multiply through by this
common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9,
and so on:

This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from the original series s cancels every term in the original but the first:

A similar technique can be used to evaluate any self-similar expression.

Formula

For , the sum of the first n terms of a geometric series is:

where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:

As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes

When a = 1, this simplifies to:

the left-hand side being a geometric series with common ratio r. We can derive this formula:

The general formula follows if we multiply through by a.
This formula is only valid for convergent series (i.e., when the magnitude of r is less than one). For example, the sum is undefined when r = 10, even though the formula gives s = −1/9.
This reasoning is also valid, with the same restrictions, for the complex case.

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:

Search This Blog

About Me

Lifetime Chartered Member" of "The Chartered Institute of Logistics & Transport - India", New Delhi . I am working a Trainer, Tutor and Course Provider for Logistics and Supply Chain Management . Also provide counseling service to students and professionals regarding education, training and job in the field of Logistics and Supply Chain Management .